Maximum likelihood estimation for the generalized poisson distribution

The generalized Poisson distribution;containing two

parameters and studied by many researchers; describes the distribution of busy periods under a queueing system and has very interesting properties; The probabilities for successive classes depend upon the previous occurrences; The problem of admissible maximum likelihood estimators for for the parameters Is discussed and a necessary and sufficient condition is derived for which unique admissible maximum likelihood estimators exist; The first; order terms in the biases; variances and the covariance of these maximum likelihood estimators are obtained.     

Moment properties of estimators for a type 1 extreme-value regression model

A regression model is considered in which the response variable has a type 1 extreme-value distribution for smallest values. Bias approximations for the maximum likelihood estimators are pivm and a bias reduction estimator for the scale parameter is proposed. The small sample moment properties of the maximum likelihood estimators are compared with the properties of the ordinary least squares estimators and the best linear unbiased estimators based on order statistics for grouped data.     

Estimation for the three-parameter inverse gaussian distribution

The three-parameter inverse Gaussian distribution is defined and moment estimators and maximum likelihood estimators are obtained. The moment estimators are found in closed form and their asymprotic normality is proven. A sufficient condition is provided for the existence of the maximum likelihood estimators.     

Estimation for Location-Scale Models with Censored Data

This article considers a class of estimators for the location and scale parameters in the location-scale model based on ‘synthetic data’ when the observations are randomly censored on the right. The asymptotic normality of the estimators is established using counting process and martingale techniques when the censoring distribution is known and unknown, respectively. In the case when the censoring distribution is known, we show that the asymptotic variances of this class of estimators depend on the data transformation and have a lower bound which is not achievable by this class of estimators. However, in the case that the censoring distribution is unknown and estimated by the Kaplan–Meier estimator, this class of estimators has the same asymptotic variance and attains the lower bound for variance for the case of known censoring distribution. This is different from censored regression analysis, where asymptotic variances depend on the data transformation. Our method has three valuable advantages over the method of maximum likelihood estimation. First, our estimators are available in a closed form and do not require an iterative algorithm. Second, simulation studies show that our estimators being moment-based are comparable to maximum likelihood estimators and outperform them when sample size is small and censoring rate is high. Third, our estimators are more robust to model misspecification than maximum likelihood estimators. Therefore, our method can serve as a competitive alternative to the method of maximum likelihood in estimation for location-scale models with censored data. A numerical example is presented to illustrate the proposed method.     

On the estimation of the parameters of the von mises distribution

The usual maximum likelihood estimators of the parameters of the von Mises distribution are shown to perform badly in small samples. In view of this and the fact that these estimators require a large amount of computation, alternative, simpler estimators are proposed. It is shown that these estimators are at least comparable to the traditional estimators and are, in many cases, superior to them. We also apply the procedure of jackknifing to the maximum likelihood estimator of the concentration parameter of the von Mises distribution and compare the properties of the jackknifed estimator with the other estimators considered in this paper.     

Minimum distance estimators in extreme value distributions

We define minimum distance estimators for the parameters of the extreme value distribution G_o based on the Cramer-von-Mises distance. These estimators are rather robust and consistent, but asymptotically less efficient than the maximum likelihood estimators which are not robust. A small simulation study for finite sample size show that under G_o the finite efficiency of the minimum distance estimators is rather similar to the maximum likelihood ones.     

Estimation for 3-parameter lognormal distribution with unknown shifted origin

A new reparameterization of a 3-parameter lognormal distribution with unknown shifted origin is presented by using a dimensionless parameter. We avoid, in this article, the application of logarithmic and exponential transformations to a value which has a physical dimension. The distribution function contains two dimensional parameters and one dimensionless parameter. Modified moment estimators and maximum likelihood estimators are presented. The presented modified moment estimators and maximum likelihood estimators are confronted with some actual data.     

Closed-Form Estimators for the Gamma Distribution Derived From Likelihood Equations

It is well-known that maximum likelihood (ML) estimators of the two parameters in a gamma distribution do not have closed forms. This poses difficulties in some applications such as real-time signal processing using low-grade processors. The gamma distribution is a special case of a generalized gamma distribution. Surprisingly, two out of the three likelihood equations of the generalized gamma distribution can be used as estimating equations for the gamma distribution, based on which simple closed-form estimators for the two gamma parameters are available. Intuitively, performance of the new estimators based on likelihood equations should be close to the ML estimators. The study consolidates this conjecture by establishing the asymptotic behaviors of the new estimators. In addition, the closed-forms enable bias-corrections to these estimators. The bias-correction significantly improves the small-sample performance.     

Classical Estimator of Parameters of the Gamma Case With Presence of Two Outliers

The authors derive the moment, maximum likelihood, and mixture estimators of parameters of the gamma distribution with presence of two outliers generated from uniform distribution. These estimators are compared empirically when all the parameters are unknown; their bias and mean squared error are investigated with the help of numerical technique. The authors shown that these estimators are asymptotically unbiased. At the end, they conclude that mixture estimators are better than the maximum likelihood and moment estimators.     

Approximate maximum likelihood estimation of the mean and standard deviation of the normal distribution based on type ii censored samples

It is known that the maximum likelihood methods does not provide explicit estimators for the mean and standard deviation of the normal distribution based on Type II censored samples. In this paper we present a simple method of deriving explicit estimators by approximating the likelihood equations appropriately. We obtain the variances and covariance of these estimators. We also show that these estimators are almost as eficient as the maximum likelihood (ML) estimators and just as eficient as the best linear unbiased (BLU), and the modified maximum likelihood (MML) estimators. Finally, we illustrate this method of estimation by applying it to Gupta's and Darwin's data.     

Estimation of three-parameter exponentiated-Weibull distribution under type-II censoring

The present article obtains the point estimators of the exponentiated-Weibull parameters when all the three parameters of the distribution are unknown. Maximum likelihood estimator generalized maximum likelihood estimator and Bayes estimators are proposed for three-parameter exponentiated-Weibull distribution when available sample is type-II censored. Independent non-informative types of priors are considered for the unknown parameters to develop generalized maximum likelihood estimator and Bayes estimators. Although the proposed estimators cannot be expressed in nice closed forms, these can be easily obtained through the use of appropriate numerical techniques. The performances of these estimators are studied on the basis of their risks, computed separately under LINEX loss and squared error loss functions through Monte-Carlo simulation technique. An example is also considered to illustrate the estimators.     

On the failure rate estimation of the inverse gaussian distribution

New estimators of the inverse Gaussian failure rate are proposed based on the maximum likelihood predictive densities derived by Yang (1999). These estimators are compared, via Monte Carlo simulation, with the usual maximum likelihood estimators of the failure rate and found to be superior in terms of bias and mean squared error. Sensitivity of the estimators against the departure from the inverse Gaussian distribution is studied.     

Estimation in a truncated bivariate poisson distribution

Moment estimators for parameters in a truncated bivariate Poisson distribution are derived in Hamdan (1972) for the special case of λ₁ = λ₂, Where λ₁, λ₂ are the marginal means. Here we derive the maximum likelihood estimators for this special case. The information matrix is also obtained which provides asymptotic covariance matrix of the maximum likelihood estimators. The asymptotic covariance matrix of moment estimators is also derived. The asymptotic efficiency of moment estimators is computed and found to be very low.     

Estimating spatial means with an application to remotely sensed data

The paper examines alternative estimators for the mean of a spatial process where observations are not independent. Properties of the sample mean and its standard error are contrasted with those of maximum likelihood estimators derived for three spatial models. The information loss caused by spatial dependency in the data is examined. The distribution theory for the estimators is reviewed and the paper concludes with an empirical example illustrating the properties of the estimators and the practical benefits of the maximum likelihood procedure.     

A novel weighted likelihood estimation with empirical Bayes flavor

We propose a novel approach to estimation, where a set of estimators of a parameter is combined into a weighted average to produce the final estimator. The weights are chosen to be proportional to the likelihood evaluated at the estimators. We investigate the method for a set of estimators obtained by using the maximum likelihood principle applied to each individual observation. The method can be viewed as a Bayesian approach with a data-driven prior distribution. We provide several examples illustrating the new method and argue for its consistency, asymptotic normality, and efficiency. We also conduct simulation studies to assess the performance of the estimators. This straightforward methodology produces consistent estimators comparable with those obtained by the maximum likelihood method. The method also approximates the distribution of the estimator through the “posterior” distribution.     

Maximum likelihood vs. maximum goodness of fit estimation of the three-parameter Weibull distribution

The use of statistics based on the empirical distribution function is analysed for estimation of the scale, shape, and location parameters of the three-parameter Weibull distribution. The resulting maximum goodness of fit (MGF) estimators are compared with their maximum likelihood counterparts. In addition to the Kolmogorov–Smirnov, Cramer–von Mises, and Anderson–Darling statistics, some related empirical distribution function statistics using different weight functions are considered. The results show that the MGF estimators of the scale and shape parameters are usually more efficient than the maximum likelihood estimators when the shape parameter is smaller than 2, particularly if the sample size is large.     

An extension of the half-normal distribution

In this paper we introduce a new distribution, namely, the slashed half-normal distribution and it can be seen as an extension of the half-normal distribution. It is shown that the resulting distribution has more kurtosis than the ordinary half-normal distribution. Moments and some properties are derived for the new distribution. Moment estimators and maximum likelihood estimators can computed using numerical procedures. Results of two real data application are reported where model fitting is implemented by using maximum likelihood estimation. The applications illustrate the better performance of the new distribution.     

Estimation of parameters of a right truncated exponential distribution

The maximum likelihood, moment and mixture of the estimators are for samples from the right truncated exponential distribution. The estimators are compared empirically when all the parameters are unknown; their bias and mean square error are investigated with the help of numerical technique. We have shown that these estimators are asymptotically unbiased. At the end, we conclude that mixture estimators are better than the maximum likelihood and moment estimators.     

Estimating the parameters of a doubly truncated normal distribution

This paper deals with the maximum likelihood estimation of parameters for a doubly truncated normal distribution when the truncation points are known. We prove, in this case, that the MLEs are nonexistent (become infinite) with positive probability. For estimators that exist with probability one, the class of Bayes modal estimators or modified maximum likelihood estimators is explored. Another useful estimating procedure, called mixed estimation, is proposed. Simulations compare the behavior of the MLEs, the modified MLEs, and the mixed estimators which reveal that the MLE, in addition to being nonexistent with positive probability, behaves poorly near the upper boundary of the interval of its existence. The modified MLEs and the mixed estimators are seen to be remarkably better than the MLE     

Modified maximum likelihood and modified moment estimators for the three-parameter weibull distribution

This article is concerned with modifications of both maximum likelihood and moment estimators for parameters of the three-parameter Wei bull distribution. Modifications presented here are basically the same as those previously proposed by the authors (1980, 1981, 1982) in connection with the lognormal and the gamma distributions. Computer programs were prepared for the practical application of these estimators and an illustrative example is included. Results of a simulation study provide insight into the sampling behavior of the new estimators and include comparisons with the traditional moment and maximum likelihood estimators. For some combinations of parameter values, some of the modified estimators considered here enjoy advantages over both moment and maximum likelihood estimators with respect to bias, variance, and/or ease of calculation.